Start t-loop by evaluating

ABSTRACT

A NOVEL METHOD FOR OPTIMALLY RECLAIMING X STORAGE IN THE 1-2-3 GNSO TECHNIQUE IS SET FORTH. THE METHOD DEFINES (1) THE MINIMAL STORAGE WHICH IS NECESSARY TO CARRY OUT THE 1-2-3 GNSO TECHNIQUE (2) DEFINES THE MINIMAL STORAGE NECESSARY TO CARRY OUT VARIABILITY-TYPE GAUSSIAN ELIMINATION BY LOOPING TYPE CODE (3) REDUCES THE AMOUNT OF CODE NECESSARY FOR LARGE PROBLEMS AS THE BASE REGISTERS NEED ONLY SPAN THE MINIMAL SET OF C VALUES (4) DEFINES THE VALUES OF THE A MATRIX WHICH CAN BE USED FOR C STORAGE (5) REALIZES EFFECTS 1-4 FOR THE CASE OF FOUR OR MORE DISTINCT VARIABILITY TYPES APPEARING AS ARGUMENTS IN THE COEFFICIENTS OF THE MATRIX A. IN CARRYING OUT THE METHOD, ASSUMING THAT STEP K OF THE OPTORD ALGORITHM, AS DESCRIBED IN THE PUBLICATION OF G.D. HACHTEL, R.K. BRAYTON AND F.G. GUSTAVSON, &#34;THE SPARSE TABLEAU APPROACH TO NETWORK ANALYSIS AND DESIGN,&#34; IEEE TRANSACTIONS ON CIRCUIT THEORY, JANUARY 1971, PP. 101-113, HAS BEEN ATTAINED WHEREBY A(K) IS HAD, THEN A PIVOT IS SELECTED AND A(K) IS UPDATED TO A (K+1) BY PERFORMING AN OUTER PRODUCT OPERATION. THE FOLLOWING RULES ARE THEN VALID (A) $ IS AVAILABE IF IT IS OF THE X TYPE AND PIVOT STEP I HAS BEEN REACHED OR IF THE LAST ELEMENT IN ROW K OF U HAS BEEN PROCESSED, I.E., LIK IS AVAILABLE AT PIVOT STEP $ WHERE 1=MIN. (I,S) AND S=MAX. ($) (B) UKJJ&gt;K IS AVAILABLE IF IT IS OF THE X TYPE AND XK IS NOT NEEDED TO SOLVE UX=Y AND WHEN PIVOT J IS REACHED OR WHEN THE LAST ELEMENT OF COLUMN K OF L IS PROCESSED, I.E., AT PIVOT STAGE J&gt;K WHERE   K=MIN. (J,S) AND S=MAX. ($)   (C) IF THE FORMULA IS NOT C=A, THEN WHEN AJJ(X) IS USED, IT CAN BE RECLAIMED, I.E., THE VALUE OF C IS STORED IN THE A SPACE.

DEFENSIVE PUBLICATION UNITED STATES PATENT OFFICE Published at the request of the applicant or owner in accordance with the Notice of Dec. 16, 1969. 869 0.G. 687. The

, abstracts of Defensive Publication applications are identified by distinctly numbered series and are arranged chronologically.

The heading of each abstract indicates the number of pages of specification, including claims and sheets of drawings contained in the application as originally filed. The files of these applications are available to the public for inspection and reproduction may be purchased for 30 cents a sheet.

' Defensive Publication applications have not been examined as to the merits of alleged invention. The Patent Office makes no assertion as to the novelty of the disclosed subject matter- PUBLISHED DECEMBER 4, 1973 T917,010 METHOD OF STORAGE RECLAMATION 1N SPARSE MATRIX TECHNIQUES Robert K. Brayton, Briarcliif Manor, and Fred G. Gustavson and Gary D. Hachtel, Ossining, N.Y., assignors to International Business Machines Corporation, Armonk,

Filed Apr. 6, 1973, Ser. No. 348,832 Int. Cl. G061? 15/32 US. Cl. 444-1 3 Sheets Drawing. 32 Pages Specification A novel method for optimally reclaiming X storage in the 1-2-3 GNSO technique is set forth'. The method defines (l) the minimal storage which is necessary to carry out the l-23 GNSO technique (2) defines the minimal storage necessary to carry out variability-type Gaussian elimination by looping type code (3) reduces the amount of code necessary for large problems as the base registers need only span the minimal set of C values (4) defines the values of the A matrix which can be used for C storage (5) realizes effects 1-4 for the case of four or more distinct variability types appearing as arguments in the coeflicients of the matrix A.

(A) l izk is available if it is of the X type and pivot step i has been reached or if the last element in row k of U has been processed, i.e., I is available at pivot step 12k where I=min. (i,s) and s=max. {l/U 0} (B) u i k is available if it is of the X type and x is not needed to solve Ux=y and when pivot i is reached or when the last element of column k of L is processed, i.e., at pivot stage j k where k=min. (]',s) and s=max. {zr/l a o} (C) if the formula is not C=A, then when a (x) is used,

it can be reclaimed, i.e., the value of C is stored in the A space.

Dec. 4, 1973 K, BRAYTON ETAL MATRIX TECHNIQUES METHOD OF STORAGE RECLAMATION IN SPARSE Filed April 6, 1973 3 Sheets-Sheet 1 FIG. 1

INITIALTZE x AND L -10 PERFORM ORDERTNC \12 BASED ON WEICHTED OPERATION COUNT COMPUTE VA=V(C(A)) AND DA=D(G(A)) 14 START t-LOOP BY EVALUATING 16 T(t)=(V(A)=1)*A CALCULATE CT=G (T,A) 18 START x-LOOP BY EVALUATING \20 X(x,t) =(VA=2)*A+(DA-2)*G (T,A)

CALCULATE GX=C2(X,A) 22 LAST x FOR THIS t 24 YES NO IF LAST t,STOP UPDATE x UPDATE t 28 Dec. 4, 1973 R K, BRAYTON ET AL T917,010

METHOD OF STORAGE RECLAMATION IN SPARSE MATRIX TECHNIQUES iled April 1973 5 Sheets-Sheet 2 INITIALIZE VA= V(A); DA=V(A) FOR k=1,2,...,N-1 EXECUTE STEPS 36 AND 58 UPDATE VA OF U FOR j=k+1,...,N. (N) (N) (k+1) (k) (k) DA =VA ,VA=VA SET vA =VA TVA rm rm v/ikPARTlAL L\U UPDATE FOR I=k 1,...,N,SET MU=0-,ML=0; +1) (k) (k) (k) (k+1) F(0RJ)-k+1,...,N, SET VAIJ- -VA -[VA -VA ,ML MLTVA k+1, (k+1) (k+1) MU uufvA m SET DAM DA vA TMu, DA vA [ML FIG. 3 m PARTIAL GAUSSIAN ELIMINATION -42 b |NITTAL|ZE:P(1)=A,,; k=1 -44 FOR k=1,2,...,N-1 EXECUTE BLOCKS 48,50, AND 52' TEST U-COMPLETION FOR j=k+T,...,N. SET (S (A A) TEST PARTIAL L\U UPDATE FOR i=k+1,...,N -50 T FOR j=k+1,...,N, SET P P Dec. 4, 1973 Filed April 6, 1973 R K. BRAYTON ET METHOD OF STORAGE RECLAMATION IN SPARSE MATRIX TECHNIQUES 3 Sheets-Sheet 3 

